Paper · Jul 23, 2025

The dynamical Alekseevskii conjecture in dimension five

Maher Billon

math.DGarXiv:2507.17781
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove the dynamical Alekseevski conjecture in dimension five. We also provide a detailed analysis of the homogeneous Ricci flows on and .

View PDF View HTML arXiv page
Submitted
Jul 23, 2025
Subjects
math.DG
Ricci flowHomogeneous spaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On the embeddability of the homogeneous Ricci flow and its collapses (Mauro Patrão, Lucas Seco and Llohann D. Sperança, 2021)5Patrão 2021 On the dynamics of positively curved metrics on SU(3)/T² under the homogeneous Ricci flow (Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins, 2023)4Cavenaghi 2023 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)3Araujo 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)1Araujo 2023 Riemannian Geometry of G₂-type Real Flag Manifolds (Brian Grajales, Gabriel Rondón and Julieth Saavedra, 2024)6Grajales 2024 Finite extinction time of a family of homogeneous Ricci flows (Roberto Araujo, 2025)2Araujo 2025 Refined behavior description of the normalized Ricci flow on homogeneous spaces (Nurlan A. Abiev, 2025)8Abiev 2025 Ricci flows and bracket flows on twisted principal bundles (Henry Wood, 2026)7Wood 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces · Shared terms “Alekseevskii Conjecture”, “Dynamical Alekseevskii”

  2. Finite extinction time of a family of homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces · Shared terms “Alekseevskii Conjecture”, “Dynamical Alekseevskii”

  3. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

    Topics Homogeneous spaces

  4. On the dynamics of positively curved metrics on under the homogeneous Ricci flow

    Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins

    Topics Homogeneous spaces

  5. On the embeddability of the homogeneous Ricci flow and its collapses

    Mauro Patrão, Lucas Seco and Llohann D. Sperança

    Topics Homogeneous spaces · Shared terms “detailed”

  6. Riemannian Geometry of -type Real Flag Manifolds

    Brian Grajales, Gabriel Rondón and Julieth Saavedra

    Topics Homogeneous spaces · Shared terms “dynamical”

  7. Ricci flows and bracket flows on twisted principal bundles

    Henry Wood

    Topics Homogeneous spaces

  8. Refined behavior description of the normalized Ricci flow on homogeneous spaces

    Nurlan A. Abiev

    Topics Homogeneous spaces

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.